If A its nxn Matrix and A^4 - A^3 + A^2 - A + Ιnxn = 0nxn, prove Α^-1 = -Α^4
"If A its nxn Matrix and A^4 - A^3 + A^2 - A + Ιnxn = 0nxn, prove Α^-1 = -Α^4"
You know the question better than me, are you supposed to assume that A is invertible? It IS possible to prove that A is invertible, so I don't think you're supposed to assume that.
First you must prove that,A is invertible because if it don't we can't prove Α^-1 = -Α^4
I know that, but I'm not going to give you the answer, I want you to try first
I think i know how to prove it im in 2nd year in Greek university in Math science but this is a lesson for 1st year
and I'm doing it again!
Its -A^4 + A^3 - A^2 + A = I <=>
<=> A( -A^3 + A^4 - A + I) =I
so ( -A^3 + A^4 - A + I)=A^-1
Yes I think you got it, But just to be sure:
You got the equation A*(-A3+A2-A1+I)=I. We can replace -A3+A2-A1+I with "B". So we got A*B=I. I is an invertible matrix, and if a multiplication of two matrices results in an invertible matrix this means that both matrices are also invertible, meaning that A is invertible.